A pendulum has a mass of 3.6 kg, a length of 2.1 meters, and swings through a (half)arc of 26.2 degrees. What is its amplitude to the nearest centimeter?
If by "amplitude" you mean the horizontal back-and-forth motion, that is L*sin26.2 = 0.927 meters = 93 cm (rounded).
There is also up-and-down motion of L*(1 - cos26.2) = 0.216 meters
The amplitude could also be expressed as the half angle, 26.2 degrees. It all depends upon how you define it.
To find the amplitude of a pendulum, we need to understand what amplitude means in the context of pendulum motion. The amplitude of a pendulum is the maximum distance the pendulum swings away from its equilibrium position.
In this case, we are given the mass of the pendulum (m = 3.6 kg), the length of the pendulum (L = 2.1 meters), and the angle through which it swings (θ = 26.2 degrees).
To find the amplitude, we can use the simple pendulum equation:
θ = amplitude × sin(ωt)
Where θ is the angle, amplitude is the maximum angular displacement from the equilibrium position, and ω is the angular frequency (ω = √(g/L)).
First, let's convert the angle from degrees to radians:
θ_radians = θ × (π/180)
θ_radians = 26.2° × (π/180)
θ_radians ≈ 0.456 radians.
Now, rearrange the equation to solve for the amplitude:
amplitude = θ_radians × L
Plug in the known values:
amplitude ≈ 0.456 radians × 2.1 meters
amplitude ≈ 0.958 meters.
To convert the amplitude to centimeters, multiply by 100:
amplitude_cm ≈ 0.958 meters × 100 cm/meter
amplitude_cm ≈ 95.8 centimeters.
Therefore, the approximate amplitude of the pendulum is 95.8 centimeters (or 0.958 meters) when rounded to the nearest centimeter.